Two sides of a rhombus $A B C D$ are parallel to the lines $y=x+2$ and $y=7 x+3$. If the diagonals o

Question

Two sides of a rhombus $A B C D$ are parallel to the lines $y=x+2$ and $y=7 x+3$. If the diagonals of the rhombus intersect at the point $(1,2)$ and the vertex $A$ is on the $y$-axis, If the possible coordinates of A are (c,d) and (e,f) find $ 2 	imes (c+d+e+f)$.

Professor PC Thomas Classes · Answer

Let the coordinates of $A$ be $(0, a)$ The diagonal $A C$ is parallel to the bisector of the angle between the lines\ $$\begin{equation*}y=x+2 	ag{2} \end{equation*}$$ (1) $y=7 x+3$ Equation of the bisectors of (1) and (2) are given by\ $\frac{x-y+2}{\sqrt{2}}= \pm \frac{7 x-y+3}{\sqrt{50}}$\ $\Rightarrow \quad 5(x-y+2)= \pm(7 x-y+3)$\ $\Rightarrow \quad 2 x+4 y-7=0 \& 12 x-6 y+13=0$\ Thus the diagonals $A C \& B D$ are parallel to these lines\ Slope of $A M=-\frac{2}{4}=\frac{a-2}{0-1} \quad 	herefore a=\frac{5}{2}$\ or $\quad \frac{-12}{-6}=\frac{a-2}{0-1} \quad 	herefore a=0$\ Hence the coordinates of $A$ are\ $\left(0, \frac{5}{2}ight)$ or $(0,0)$\